Muscle Dynamics, Size Principle, and Stability - Research

Engineering, The Ohio State University, Columbus, OH 43210. B. T. Stokes is with the ..... This means if Ti for larger motor units is higher than Ti for smaller motor ...
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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. BME-31, NO. 7, JULY 1984

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Muscle Dynamics, Size Principle, and Stability YUAN-FANG ZHENG, STUDENT MEMBER, IEEE, HOOSHANG HEMAMI, MEMBER, IEEE, BRADFORD T. STOKES

Abstract-A mathematical model of skeletal muscle activation during small or isometric movements is discussed with the following attributes. A direct relationship between the stimulus rate and the active state is established. A piecewise linear relation between the length of the contractile element and the isometric force is considered. Hill's characteristic equation is maintained for determining the actual output force during different shortening velocities. A physical threshold model is proposed for recruitment which encompasses the size principle and its manifestations and also exceptions to the size principle. Finally, the role of spindle feedback in stability of the model is demonstrated by studying a pair of muscles.

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I. INTRODUCTION

T HE complex behavior and complicated physiological structure of skeletal muscle has prompted decades of study to define these parameters. Much remains to be known about the actual energy conversion, bioenergetics, and dynamics of the muscle. In spite of this gap in knowledge, models of the skeletal muscle, that have input-output and behavior similar to an actual muscle, are desirable. Such models would be indispensable in any quantitative studies of the behavior of the skeletal and the nervous systems by mathematical methods and analog and digital computer simulations. In many locomotion studies [1], ideal torque generators are employed at the joints to represent the action of groups of muscles. This idealization ignores muscle characteristics and the system cannot predict actual human locomotion very well. Further, the inputs to such torque generators do not correspond to valid physiological inputs to the human muscle from the nervous system. For prosthetic and orthotic devices, eventually quiet, efficient, and cosmetically appealing muscle-like force actuators are desirable that could interface with signals from the nervous systems, and would also provide sensory signals acceptable to the central nervous system. Besides, muscle-like actuators are distributed along the peripheral links, while torquegenerator actuators are concentrated at the joints, and would make the joints very heavy and cumbersome. All these issues necessitate postulation of meaningful muscle models. Since the work of Hill [2] -[41 and Wilkie [5], the mechanical behavior of active muscles has been described by a contractile component (CC) in series with a noncontractile series elastic Manuscript received May 21, 1982; revised June 23, 1983. This work was supported in part by the National Science Foundation under Grant ECS 820-1240 and in part by the Department of Electrical Engineering, The Ohio State University, Columbus, OH 43210. Y.-F. Zheng and H. Hemami are with the Department of Electrical Engineering, The Ohio State University, Columbus, OH 43210. B. T. Stokes is with the Department of Physiology, The Ohio State University, Columbus, OH 43210.

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Fig. 1. Structural model of muscle fibers.

component (SEC) [Fig. 1(a)]. For a long time, the study of muscles was concentrated on the mechanical properties of the series elastic element [6] , and the parallel elastic element [Fig. 1(b)] whose nonlinear behavior can be described by reasonably accurate functions [7]. In some further studies, the force output of the contractile element is assumed to be a function of an input from the central nervous system [8], [9]. Recently, Hatze [7], [10] has proposed a control model of the muscle based on a set of five nonlinear first-order differential equations. This model would produce a simulation system with a very large dimension. For the model presented in this paper, a threshold theory and a corresponding physical implementation are proposed that encompass the size principle and its physiological manifestations, as well as exceptions to it [11] -[14]. The role of some of the afferent fibers in stability of this model are also studied, and some of the muscle input-output attributes are studied by digital computer simulation. In Section II, a single motor unit is studied. In Section III, a recruitment and thresholding model are proposed via a physical circuit. In Section IV, the dynamics of a whole muscle are discussed and some digital computer simulations are presented. The stability analysis is carried out in Section V. All signal intensities are represented by amplitudes in order to simplify the presentation. II. A SINGLE MOTOR UNIT It is generally accepted that the muscle fiber is composed of a series arrangement of repeating structure, the sarcomeres, that extend from Z-disk to Z-disk. Contained in this repeating structure is a controllable elementary contractile unit. Also, there are series elastic elements and parallel elastic elements within each basic unit. Although the structure of these elastic elements is not exactly known [15], the experimental phenom-

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C 1984 IEEE

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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. BME-31, NO. 7, JULY 1984

ena strongly suggest their existence. The arrangement of where a1, a2, b1, b2 are appropriate constants. Now for difthese three elements can be seen in Fig. 1. Because the struc- ferent muscles the stimulus rate which excites the maximum ture is repeated in series in each muscle fiber, this arrangement tension is different, and defining a relative active state is more is also valid as a lumped model for the entire fiber. Muscle convenient. fibers normally contract in groups, referred to as muscle units It is known t-hat in response to the stimulation of the nerves, [161. The muscle fibers within a single muscle unit are quite muscles need a certain amount of time to contract and differ homogeneous, even though they may be spread throughout widely in their speed of contraction. For example, the extrathe muscle. Therefore, the physical arrangement of Fig. I ocular muscles that rotate the eyeball require only about 7.5 should be good for a muscle unit. The muscle unit, together ms, but the soleus muscle, a slow antigravity muscle in the with its associated motorneuron, is known as a motor unit hind limb, requires about 100 ms to reach its maximal tension [16]. The modeling, based on a single muscle fiber, could [14]. Taking into account the above property of the muscle also be extended to a motor unit [9], [10]. contraction, the following first-order differential equation is According to [7], the force developed by the contractile adequate to relate the active state q to Q component is the product of the active state q (the relative 4q+cq=cQ(r)A 0.Q.1 (4) amount of Ca++ bound to troponin), the length-tension funcwhere A is the maximum tension a single motor unit can tion k, and the velocity-tension function g develop and Q is the "desired" active state of the single motor f=qkg. (1) unit while q is the actual active state, and the final values of In [7], q is a function of the length of the contractile element both in (4) are the same, i.e., the actual active state reaches the desired one after some time. and the concentration y of Ca++ Now each stimulation impulse from the nervous system q = q(y, ) (2) elicits a single twitch lasting for a fraction of a second. SucThe dependence of the active state q in (2) on t could be re- cessive twitches may add up to produce a stronger action, i.e., garded as redundant, because in (1), the force is already length a partially or fully fused tetanus. If the stimulus rate is not dependent through the tension-length function k. This redun- high enough, a single motor unit shows an unfused response dancy arises from the definition of the active state. As is well (Fig. 2). If the stimulus reaches its critical frequency, the known, each basic contractile element, the sarcomere, is sur- successive contractions fuse together and cannot be distinrounded by a network called sarcoplasmic reticulum. The guished from one another [21]. But any kind of muscle is action potential is conducted through the T-system located at composed of a number of motor units and the stimulation the borders of two adjacent networks. The depolarization of could be assumed to be distributed over motor units, instead the T-system caused by the action potential depolarizes the of being synchronized. For distributed stimulation, even at membrane of the reticulum at the triad. This, in turn, triggers a low rate, a smooth contraction can be demonstrated [19]. the release of Ca"+ ions from this area [17]. The action poten- This is the reason why the behavior of a muscle usually shows tial becomes smaller towards the interior of the membrane of smooth motion. Based on this assumption, (3) and (4) are the T-system as a function of T-system electrical capacitance. acceptable and simplify the model a great deal. The length-tension function k could be derived from wellWith the shortening of the muscle, the cross-section area inknown experimental results by Gordon et al. [22] and by the creases [9], [10]. The longer the diameter of the muscle sliding filament theory of Huxley and Niedergerke [23]. This fibers, the less Ca"+ ions could be released; therefore, the relarelation has been approximated by a mathematical expression tive amount of Ca"+ bound to troponin depends on the muscle similar to a step response of a underdamped second-order length. But according to Hill [4], the active state is defined as equation [7]: the tension developed when the contractile component is k() = 0.32 + 0.71 exp {-1.1 12( - 1)1 neither lengthening nor shortening. With this definition, the active state could be measured by the produced isometric X sin {3 - 722( - 1)} 0.5 8 < t 6 1 .8. (5) tension [18]. The relation between free calcium ion concenFig. 3 shows a comparison of k from (5) to others obtained tration and contractile response has been studied. Hellam and Podolsky [19] were the first to use skinned fibers for the from experiments [22]. From Fig. 3, the tension-length relaquantitative study of this relation. They found that the ten- tion can be represented by piecewise linear function sion was related to the free Ca++ ion concentration via a 0 0